Diffusion-limited adaptive battery charging

ABSTRACT

Some embodiments of the present invention provide a system that adaptively charges a battery, wherein the battery is a lithium-ion battery which includes a transport-limiting electrode governed by diffusion, an electrolyte separator and a non-transport-limiting electrode. During operation, the system determines a lithium surface concentration at an interface between the transport-limiting electrode and the electrolyte separator based on a diffusion time for lithium in the transport-limiting electrode. Next, the system calculates a charging current or a charging voltage for the battery based on the determined lithium surface concentration. Finally, the system applies the charging current or the charging voltage to the battery.

RELATED APPLICATIONS

This application is a continuation application of, and hereby claimspriority under 35 U.S.C. §120 to, pending U.S. patent application Ser.No. 12/242,641, entitled “Diffusion-Limited Adaptive Battery Charging,”by inventors Thomas C. Greening, P. Jeffrey Ungar and William C. Athas,filed on 30 Sep. 2008. U.S. Ser. No. 12/242,641 claims priority under 35U.S.C. §119 to U.S. Provisional Patent Application No. 61/044,160 filed11 Apr. 2008, entitled “Diffusion-Limited Adaptive Charging,” byinventors Thomas C. Greening, P. Jeffrey Ungar and William C. Athas.

BACKGROUND

1. Field

The present invention generally relates to techniques for charging abattery. More specifically, the present invention relates to a methodand apparatus for charging a lithium-ion battery which adaptivelycontrols the lithium surface concentration to remain within set limits.

2. Related Art

Rechargeable lithium-ion batteries are presently used to provide powerin a wide variety of systems, including laptop computers, cordless powertools and electric vehicles. FIG. 1 illustrates a typical lithium-ionbattery cell, which includes a porous graphite electrode, a polymerseparator impregnated with electrolyte, and a porous cobalt dioxideelectrode. The details of the transport of lithium and lithium ions inand out of the electrode granules and through the material between themare complex, but the net effect is dominated by slow diffusion processesfor filling one electrode with lithium while removing it from the other.

Note that FIG. 1 provides a physical model for the layout of a typicallithium-ion cell, wherein the oxidation and reduction processes thatoccur during charging are also illustrated. The physical model shows thecurrent collectors, which are in turn connected to the batteryterminals; the polymer separator; and the positive and negative porouselectrodes. Note that an electrolyte permeates the porous electrodes andthe separator.

The negative electrode includes granules of graphite held together witha conductive binder (in practice, there may also be a nonconductivebinder). Surrounding each graphite particle is a thin passivating layercalled the solid-electrolyte interphase (SEI) that forms when a freshcell is charged for the first time from the lithium atoms in thegraphite reacting directly with the electrolyte. This occurs because thetendency for the lithium atoms to remain in the graphite is relativelyweak when the cell is fully charged, but after the SEI is formed, theSEI acts as a barrier against further reactions with the electrolyte.Nevertheless, the SEI still allows transport of lithium ions, albeitwith some degree of extra resistance.

The positive electrode includes granules of lithiated cobalt dioxideheld together with binders similar to the negative electrode. AnySEI-like layer surrounding these particles is likely to be of much lesssignificance than in the negative electrode because lithium atomsstrongly favor remaining in these particles rather than leaving andreacting directly with the electrolyte.

Lithium transport in the negative graphite electrode (also referred toas the “transport-limiting electrode”) is slower than in the positivecobalt dioxide electrode (also referred to as the“non-transport-limiting electrode”), and therefore limits the maximalspeed of charging. During charging, the slow diffusion causes atransient build-up of lithium on the surfaces of the graphite thatvaries in direct proportion to the charging current and a characteristicdiffusion time.

The diffusion time is typically on the order of hours and has a strongdependence on temperature and other variables. For instance, a cell at15° C. can have a diffusion time which is ten times slower than a cellat 35° C. The diffusion time can also vary significantly between cells,even under the same environmental conditions, due to manufacturingvariability.

If the concentration of lithium at the surface reaches the saturationconcentration for lithium in graphite, more lithium is prevented fromentering the graphite electrode until the concentration decreases. Aprimary goal of conventional battery-charging techniques is to avoidlithium surface saturation, while keeping the charging time to aminimum. For example, one conventional technique charges at a constantcurrent until a fixed upper voltage limit (e.g., 4.2 V) is reached, andthen charges by holding the voltage constant at this upper limit untilthe current tapers to some lower limit. Note that it is common practiceto express all currents in terms of the cell capacity. For example, fora cell with a capacity of Qmax=2500 mA·hr, a “1 C” current would be 2500mA. In these units, the constant current charging is usually done atless than 1 C (e.g., 0.3 C), and the constant voltage phase isterminated when the current tapers to some value less than 0.05 C.

FIG. 2 illustrates a representative conventional charging profile. Theproblem with a conventional charging scheme is that it largely operatesblindly; the only information used is the cell voltage, which does notdirectly correlate to the lithium surface concentration. Consequently,conventional charging both misses the opportunity to use more currentwhen it is possible to do so, and enters the saturation region iflithium transport is slower than expected.

Hence, what is needed is a method and an apparatus for charging alithium-ion battery that does not suffer from the drawbacks of theseexisting techniques.

SUMMARY

Some embodiments of the present invention provide a system thatadaptively charges a battery, wherein the battery is a lithium-ionbattery which includes a transport-limiting electrode, an electrolyteseparator and a non-transport-limiting electrode. To charge the battery,the system first determines a lithium surface concentration at aninterface between the transport-limiting electrode and the electrolyteseparator. Next, the system uses the determined lithium surfaceconcentration to control a charging process for the battery so that thecharging process maintains the lithium surface concentration within setlimits.

In some embodiments, determining the lithium surface concentrationinvolves determining a potential of the transport-limiting electrodewith respect to a known reference, wherein the potential is correlatedwith the lithium surface concentration. In these embodiments, using thedetermined lithium surface concentration to control the charging processinvolves using the determined potential of the transport-limitingelectrode in a control loop, which adjusts either a charging voltage ora charging current, to maintain the potential of the transport-limitingelectrode at a level which keeps the lithium surface concentrationwithin the set limits.

In some embodiments, maintaining the potential of the transport-limitingelectrode involves maintaining a minimum potential or a maximumpotential which keeps the lithium surface concentration within the setlimits. For example, for a negative electrode, the lithium surfaceconcentration can be maintained below a saturation level, whereas for apositive electrode, the lithium surface concentration can be maintainedabove a depletion value. (Note that the term “set limits” as used inthis specification and the appended claims refers to one or more setlimits.)

In some embodiments, determining the potential of the transport-limitingelectrode involves directly measuring the potential of thetransport-limiting electrode.

In some embodiments, determining the potential of the transport-limitingelectrode involves: determining a state of charge for the battery; anddetermining the potential of the transport-limiting electrode from thedetermined state of charge and other parameters related to the battery.

In some embodiments, determining the potential of the transport-limitingelectrode involves: monitoring a temperature of the battery; monitoringa current through the battery; monitoring a total cell voltage of thebattery; and determining the potential of the transport-limitingelectrode based on the monitored temperature, current and total cellvoltage.

In some embodiments, the transport-limiting electrode is a negativeelectrode, and the non-transport-limiting electrode is a positiveelectrode.

In some embodiments, the negative electrode is comprised of graphiteand/or TiS₂; the electrolyte separator is a liquid electrolyte comprisedof LiPF₆, LiBF₄ and/or LiClO₄ and an organic solvent; and the positiveelectrode is comprised of LiCoO₂, LiMnO₂, LiFePO₄ and/or Li₂FePO₄F.

In some embodiments, determining the lithium surface concentrationinvolves: measuring a diffusion time τ for lithium in thetransport-limiting electrode; and estimating the lithium surfaceconcentration between τ measurements based on the diffusion time τ, acell capacity Q_(max) for the battery and a measured charging current Ifor the battery.

In some embodiments, measuring the diffusion time τ involvesperiodically performing a sequence of operations, including: (1)charging the battery with a fixed current for a fixed time period; (2)entering a zero current state in which the charging current is set tozero; (3) during the zero current state, measuring an open circuitvoltage for the battery at two times while the open circuit voltagerelaxes toward a steady state; and (4) calculating the diffusion time τbased on the measured open circuit voltages.

Some embodiments of the present invention provide a system thatadaptively charges a battery, wherein the battery is a lithium-ionbattery which includes a transport-limiting electrode, an electrolyteseparator and a non-transport-limiting electrode. To charge the battery,the system monitors a current through the battery, a voltage of thebattery, and a temperature of the battery. The system then uses themonitored current, voltage and temperature to control a charging processfor the battery so that the charging process maintains a lithium surfaceconcentration at an interface between the transport-limiting electrodeand the electrolyte separator within set limits.

One embodiment of the present invention provides a battery with anadaptive charging mechanism. This battery includes a transport-limitingelectrode, an electrolyte separator, and a non-transport-limitingelectrode. It also includes a current sensor to measure a chargingcurrent for the battery, and a voltage sensor to measure a voltageacross terminals of the battery. The battery additionally includes acharging source configured to apply a charging current or a chargingvoltage to the battery. This charging source operates under control of acontroller, which receives inputs from the current sensor and thevoltage sensor and sends a control signal to the charging source. Duringthe charging process, the controller controls the charging source tomaintain a lithium surface concentration at an interface between thetransport-limiting electrode and the electrolyte separator within setlimits.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 illustrates a lithium-ion battery in accordance with anembodiment of the present invention.

FIG. 2 illustrates a conventional charging profile for a lithium-ionbattery.

FIG. 3 illustrates a lumped representation of a lithium-ion cell.

FIG. 4 illustrates a relaxed open-circuit voltage versus state of chargefor a lithium-ion cell.

FIG. 5 provides a schematic representation of a Cole-Cole impedance plotfor a typical lithium-ion cell.

FIG. 6 illustrates an equivalent circuit which captures most of thefeatures in the impedance plot in FIG. 5.

FIG. 7 illustrates a lithium concentration profile.

FIG. 8 illustrates the lithium concentration through the graphite inresponse to a constant current.

FIG. 9 illustrates a relaxed open-circuit voltage for a lithium-ion cellversus the state of charge in accordance with an embodiment of thepresent invention.

FIG. 10 illustrates a temperature correction factor as a function of astate of charge in accordance with an embodiment of the presentinvention.

FIG. 11 illustrates sample data for cell voltage versus time for a pulsedischarge in accordance with an embodiment of the present invention.

FIG. 12 illustrates relaxation after a current discharge pulse inaccordance with an embodiment of the present invention.

FIG. 13 illustrates diffusion times derived from relaxation afterdischarge in accordance with an embodiment of the present invention.

FIG. 14 illustrates ideal charging with a constant diffusion time inaccordance with an embodiment of the present invention.

FIG. 15 illustrates ideal charging with a varying diffusion time inaccordance with an embodiment of the present invention.

FIG. 16 illustrates simulation results for charging a cell usingdiffusion-limited adaptive charging in accordance with an embodiment ofthe present invention.

FIG. 17 illustrates a rechargeable battery that supports adaptivecharging in accordance with an embodiment of the present invention.

FIG. 18 presents a flow chart illustrating the charging process inaccordance with an embodiment of the present invention.

FIG. 19 presents another flow chart illustrating the charging process inaccordance with an embodiment of the present invention.

FIG. 20 presents a flow chart illustrating the process of determining avoltage across a transport-limiting electrode in accordance with anembodiment of the present invention.

FIG. 21 presents a flow chart illustrating a charging process based onmeasuring the diffusion time τ in accordance with an embodiment of thepresent invention.

FIG. 22 presents a flow chart illustrating the process of measuring thediffusion time τ in accordance with an embodiment of the presentinvention.

DETAILED DESCRIPTION

The following description is presented to enable any person skilled inthe art to make and use the invention, and is provided in the context ofa particular application and its requirements. Various modifications tothe disclosed embodiments will be readily apparent to those skilled inthe art, and the general principles defined herein may be applied toother embodiments and applications without departing from the spirit andscope of the present invention. Thus, the present invention is notlimited to the embodiments shown, but is to be accorded the widest scopeconsistent with the principles and features disclosed herein.

The data structures and code described in this detailed description aretypically stored on a computer-readable storage medium, which may be anydevice or medium that can store code and/or data for use by a computersystem. The computer-readable storage medium includes, but is notlimited to, volatile memory, non-volatile memory, magnetic and opticalstorage devices such as disk drives, magnetic tape, CDs (compact discs),DVDs (digital versatile discs or digital video discs), or other mediacapable of storing computer-readable media now known or later developed.

The methods and processes described in the detailed description sectioncan be embodied as code and/or data, which can be stored in a computerreadable storage medium as described above. When a computer system readsand executes the code and/or data stored on the computer-readablestorage medium, the computer system performs the methods and processesembodied as data structures and code and stored within thecomputer-readable storage medium. Furthermore, the methods and processesdescribed below can be included in hardware modules. For example, thehardware modules can include, but are not limited to,application-specific integrated circuit (ASIC) chips, field-programmablegate arrays (FPGAs), and other programmable-logic devices now known orlater developed. When the hardware modules are activated, the hardwaremodules perform the methods and processes included within the hardwaremodules.

Adaptive Surface Concentration Charging

FIG. 3 shows a lumped element model of a cell corresponding to thephysical model shown earlier in FIG. 1. The model has a discrete elementfor the electrolyte permeating the separator and different elements forthe electrolyte permeating the two porous electrodes. The electrolytetransport properties are different in these three regions, and generallytransport of ions is expected to be fast through the separator andslower in the porous media. The graphite and cobalt dioxide lithium“insertion” materials are also represented by sets of discrete elementsat various depths into their respective electrodes. They areelectrically connected to the current collectors through successivelygreater series resistance to conduction through the binders and theinsertion materials themselves.

Lithium transport in the graphite and cobalt dioxide grains is throughdiffusion, although there can be additional rate effects resulting fromthe growth of stoichiometric phases, like LiC₁₂. The lumped model alsoshows an element for the SEI layer in series with each graphite elementsince it impedes the flow of lithium ions significantly, but stillallows the charge transfer reactions to occur. Finally, thekin/±elements account for the kinetics of the redox reactions occurringat the electrode-electrolyte interfaces. If these reactions are drivenwith currents near or exceeding their natural rates, a significantoverpotential that opposes the flow may develop.

The relaxed, open circuit voltage (relaxed OCV) across the cell dependsonly on the steady-state electrochemical reactions occurring at the twoelectrodes. Because no net current flows, there are no nonzero potentialdifferences across any of the electrolyte, SEI, kinetic, or resistiveelements in FIG. 3. Furthermore, the concentration of lithium is uniformwithin each of the electrode insertion materials, and the concentrationof Li+ ions is uniform throughout the electrolyte. Charging ordischarging the cell results in a net reversible transfer of lithiumfrom the cobalt dioxide in the positive electrode to the graphite in thenegative electrode or vice versa, respectively. Therefore, a cell'srelaxed OCV depends only on its state of charge, and, to a small extent,on temperature.

The electrochemical processes at the negative and positive electrodescan be described in terms of the respective half-cell reactionsLi_(x)C₆→Li_(x-y)C₆ +yLi⁺ +ye ⁻  (1)andLiCoO₂→Li_(1-y)CoO₂ +yLi⁺ +ye ⁻  (2)The relaxed OCV is the electrochemical potential for the full cell; thatis, ε=ε₊−ε⁻, where ε₊ and ε⁻ are the electrochemical potentials for thetwo half-cell reactions. It is convenient to use a metallic lithiumelectrode in the same electrolyte as the zero of potential; that is, touse the processLi→Li⁺ +e ⁻  (3)as a reference for the working electrode potentials. In the field, thisis commonly specified by referring to the potentials “vs. Li/Li⁺”.

In practice, such a reference is incorporated into a cell builtspecifically for testing. The reference must be in contact with theelectrolyte between the working electrodes, but insulated from directcontact with them, as shown schematically by the triangular element atthe bottom of FIG. 3.

FIG. 4 shows typical single electrode potentials vs. Li/Li⁺ as afunction of the cell's state of charge as measured using an incorporatedmetallic lithium reference electrode. The difference gives the full cellrelaxed OCV curve, which is also shown. The potential for the negativeelectrode is of particular importance, since it is a fairly low valuethrough much of the range and begins a steep decline toward zero at thetop of charge. A negative electrode potential of 0V vs. Li/Li⁺ means thelithium in the graphite is in equilibrium with metallic lithium in theelectrolyte; that is, the graphite is saturated with lithium.

A rough extrapolation of the plots in FIG. 4 shows that this cell wouldlikely saturate the graphite at a relaxed OCV near 4.24 V, which is amodest margin of safety with respect to charging the cell to 4.20 V. Fora given cell formulation, this margin is mostly affected by the balancein total electrode capacities. For example, using a thicker layer ofgraphite will stretch out the negative electrode potential curverelative to the one for the positive electrode. Therefore, at the top ofcharge (OCV of 4.20 V), the potentials for both electrodes will begreater. This is desirable for the negative electrode; the increasedpotential on the positive electrode may also be acceptable.

Thus far, we have only considered the static properties of the cell butcharging (and discharging) involves a net current flow, so some of thetransport dynamics captured by the model in FIG. 3 come into play. If,as is usually the case for this type of cell, the kinetics of theelectrochemical reactions are fast, and the other sources of internalimpedance are small, the dominant contribution to the cell voltage isstill the difference between the electrochemical potentials for the twoelectrodes. However, instead of referencing them to an overall state ofcharge for the cell, one must use an effective state of charge for eachelectrode that corresponds to the surface concentration of lithium inthe electrode's solid phase nearest the cell separator. Referring toFIG. 3, this is the potential difference between A and B, where weassume the voltage drops across the resistive elements, thekin/±elements, the SEI, and the electrolyte/separator element are small.There are a few subtleties here that merit further discussion.

First, the reason is that the lithium surface concentration in the solidis relevant is because the redox reactions all take place in the narrowregion near the solid electrolyte interface. For example, for thenegative electrode, the local electrochemical contribution to thepotential comes from the process in Equation 1, where there is chargetransfer resulting in lithium leaving the solid near the surface aslithium ions in the local electrolyte. The potential for this process isshown in the middle plot in FIG. 4, where the reference is now taken, inprinciple, to be lithium metal in electrolyte at the localconcentration. The same applies to the local contribution to thepotential from the process in Equation 2 for the positive electrode.Whether or not the surface concentration of lithium in the solid is muchdifferent from its value in the bulk depends on the current density atthe interfaces and on the rate of solid state diffusion of lithium inthe granules, but this results in no additional contribution to the cellvoltage.

Second, the regions of the electrodes nearest the separator are ofspecial importance because this is where the largest deviations fromequilibrium lithium concentration occur and where the contribution tothe cell voltage from electrolyte transport effects is smallest. Theimpedance of the electrolyte/±elements is expected to be significantlylarger than for the electrolyte in the separator since the ions musttravel through the tortuous paths formed by the pore spaces.Consequently, these will incur significant voltage drops, which in turnmeans that the local current densities and the deviations of theelectrochemical potentials from equilibrium decrease with increasingdistance from the separator. To put it another way, the net potentialthroughout either composite, porous electrode with respect to thelithium reference shown in FIG. 3 is constant and equal to its valuenearest the separator, but an increasing share of it is attributable tothe electrolyte in the pores.

When charging the cell, lithium will tend to pile up near the surfacesof the graphite granules and to deplete near the surfaces of the cobaltdioxide granules. Both electrodes will appear to be at higher states ofcharge than when equilibrated. However, they will not necessarily appearto be at the same higher state of charge; the electrode with the slowertransport, here the negative electrode, will exhibit the largerdifference.

The primary goal of an efficient adaptive charging technique is tocharge at a rate where the lithium surface concentrations, and hence theelectrode potentials, are kept within desirable limits, but as close tothem as can be managed reliably. For example, with the negativeelectrode being the limiting factor, keeping its potential too far from0V vs. Li/Li⁺ charges unnecessarily slowly, but getting too closeinvites lithium saturation in the graphite. In principle, meeting thisgoal would be very simple for a cell with a reference electrode, sinceone could simply adjust the charger to servo the negative electrodepotential to some positive value that gives a margin for error, such as50 mV vs. Li/Li⁺. This is the essence of the new Adaptive SurfaceConcentration Charging (ASCC) method.

There would, in fact, be more margin than the 50 mV because, even usinga reference electrode, we see from FIG. 3 there will be a number ofother voltage drops that will reduce the measured negative electrodepotential, and some of these have nothing to do with its tendency tosaturate. (This will be discussed in more detail in the next section.)

Without an integrated reference electrode, we can implement ASCCindirectly if we can track the cell's state of charge with sufficientaccuracy. Let us assume the state of charge of the cell is q and we wishto target a voltage for the negative electrode of v_(target). Referringto FIG. 4, and following the arguments already presented, we see thatthe voltage on the negative electrode vs. Li/Li⁺ will satisfyvC6≧ε₊(q)−v _(cell)  (4)If the positive electrode transport limitations and resistive drops wereeliminated, this lower bound would become an equality and exactly thesame as what would be measured using a reference electrode. We willsafely approach the target ifε₊(q)−v _(cell) ≧v _(target)  (5)or equivalentlyv _(cell)≦ε₊(q)−v _(target).The one issue that can arise is that the estimate can be tooconservative. In particular, any series resistance will increase themeasured v_(cell) and needlessly reduce the lower bound for the negativeelectrode voltage in Equation 4. Addressing these drops is the topic ofthe next section.Resistive Potential Correction

Referring once again to FIG. 3, if we follow the path through theresistive elements, the separator, and the electrode elements closest tothe separator, we can express the voltage across the cell at a state ofcharge q asv _(cell)=ε₊(q+Δq ₊)−ε⁻(q+Δq ⁻)+v _(kin/+) +v _(kin/−) +v _(SEI) +v_(seperator) +v _(r)  (6)

The lithium surface concentrations for the positive and negativeelectrodes have been expressed relative to the base state of charge viathe respective deviations Δq±. The v_(r) drop is the total for the paththrough the resistive elements, and the other terms correspond to theindividual elements in the diagram for the SEI and the reactionkinetics. The graphite-related voltage to manage while charging isv _(C6)≧ε⁻(q+Δq ⁻)−v _(kin/−) −v _(SEI)  (7)since if this reaches zero, the graphite will saturate at the surfaceand SEI growth may occur. If there are no limitations for the positiveelectrode, then all other impedance effects may be safely removed toestimate this voltage. Equations 6 and 7 givev _(C6)=ε₊(q+Δq ₊)+v _(kin/+) +v _(separator) +v _(r) −v _(cell).  (8)Assuming transport through the positive electrode is fast, we canneglect Δq₊ and v_(kin/+), and an improved lower bound for the negativeelectrode voltage isv _(C6)≧ε₊(q)−(v _(cell) −v _(separator) −v _(r)).  (9)If we can estimate the potential drop attributed here to the separatorand electronic conduction through the solids, then we can account forthem in an implementation of ASCC to reduce further the time to chargethe cell. An examination of the cell's electrical characteristicsprovides just such an estimate, which we now discuss.

Electrochemical Impedance Spectroscopy (EIS) is the measurement of thesmall signal, differential impedance of a cell as a function offrequency. In concept, and in one common measurement configuration, acell under test is brought to a known state of charge, and a smallsinusoidal voltage of fixed frequency is applied in superposition withthe relaxed OCV. The resulting current is measured after any transientbehavior has decayed, and is compared in magnitude and phase with theapplied voltage to give the complex impedance for this frequency. A plotof the real and imaginary parts of the impedance as the frequency isswept from low to high can reveal much about transport throughout thecell and interfacial processes at the electrodes.

FIG. 5 is a schematic representation of an impedance spectrum for atypical lithium-ion cell, also known as a Cole-Cole impedance plot. Thedifferent features in the plot correspond to different processes in thecell. For example, the 45° straight section at low frequencies ischaracteristic of a diffusive process, such as for transport of ionsthrough the electrolyte in the porous electrodes, or of lithium into thesolids, and the turn up the −Z_(imag) axis at the lowest frequenciescorresponds to the cell charging and discharging. The nearlysemicircular hump at higher frequencies is more interesting, since itlikely corresponds to the charge transfer process across the SEI film inthe negative electrode. This feature can be described roughly by aparallel RC circuit, with resistance R_(ct), as shown on the plot.

At higher frequencies still, we see the suggestion of another featuredistorting the semicircle, and finally the impedance turns sharply downthe −Z_(imag) axis, which is inductive behavior. There is a significantresidual resistive component, R_(ohmic), and this is what we can use toprovide the improved negative electrode voltage estimate suggested byEquation 9. One possible equivalent circuit that captures most of theimpedance plot features is shown in FIG. 6. The charge transfer throughthe SEI is represented by R_(ct) and a double-layer capacitance C_(dl)for the charging of the interface. The D element represents diffusivetransport in the porous electrode. R_(ohmic) is in series with theelectrode and can be taken as the solution resistance plus any otherresistance to conduction through the solids. Therefore, the potentialdrop in Equation 9 for a charging current I isv _(separator) +v _(R) =IR _(ohmic),  (10)and the lower bound for the negative electrode voltage becomesv _(C6)=ε₊(q)−(v _(cell) −IR _(ohmic)).  (11)This estimate is the basis of a practical implementation of ASCC, whichwe discuss in the next section. R_(ohmic) itself may be measured withoutdoing a full EIS scan by measuring the real part of the impedance atfrequencies high enough to “short-out” R_(ct) via C_(dl)+D in parallel.For these cells, a frequency of 1 kHz is suitable.Servo Control

In some embodiments, Adaptive Surface Concentration Charging uses aproportional-integral-derivative (PID) controller to servo the estimatedlithium surface concentration to a level below saturation by adjustingthe battery charger. Charging terminates when the current drops below agiven threshold and the cell voltage is close to the target cellvoltage.

Instead of servoing surface concentration, the estimated voltage acrossthe graphite electrode, v_(C6) from Equation 11, is servoed to a targetvoltage, v_(target). Ideally, the target voltage would be 0V at the edgeof saturation, where the lithium in graphite is in equilibrium with puremetallic lithium. To be conservative, however, the target voltage istypically set slightly higher, for instance 50 mV, to ensure that aslight overshoot in the servo, charger inaccuracy, or other errors donot cause saturation. The servo input error ε(t) is given by:ε(t)=v _(target) −v _(C6)(t),  (12)where the error is updated frequently, such as once per second.

In a multi-cell system where cells are placed in series, the estimatedgraphite electrode potential is required to be separately calculated foreach group of cells in series. Cells in parallel form a single bank,which is effectively a single cell, and cannot be treated separately. Tocharge conservatively, the most negative error ε(t) of all of the cellbanks is servoed to zero.ε_(min)(ε_(A),ε_(B), . . . )  (13)

The servo output for the PID controller in this case is the chargervoltage, although the charging current could be controlled similarly.V _(charger)(t)=K _(P)·ε_(min)(t)+K _(I)∫ε_(min)(t)dt+K _(D)(dε_(min)(t)/dt),  (14)where K_(P) is the proportional gain, K_(I) is the integral gain, andK_(D) is the derivative gain.

When using a non-zero integral gain K_(I), special considerations arerequired for the integral term initialization and prevention of integralwind-up when the output is limited. A logical initial integral termvalue would be the measured battery pack open circuit voltage V_(pack),so that the charger begins in a state with zero initial chargingcurrent. If the charger has a current limit, it is possible for theservo to set a voltage that cannot be achieved. To prevent integralwind-up, the integral term should be suspended if the current is limitedby the charger. For systems where it is difficult to know precisely thatthe charger limit has been reached, the integral term could be resumedif the servo input error were ever negative.

To prevent lithium saturation, the estimated graphite electrode voltage,v_(C6), should always be greater than 0 V; therefore, it is criticalthat the PID controller gains are tuned to prevent overshoot. Sincev_(C6) changes slowly and controller overshoot should be avoided, thereis no need to include a derivative term (K_(D)=0). The controller isthus simplified to a proportional-integral (PI) controller.

Charging is terminated when the measured battery pack voltage, V_(pack),is within a threshold, V_(thresh), of the desired charging voltage,V_(charge), and the current drops below a minimum charging currentlevel, I_(min), for a time of at least t_(termination).

Battery Charging Based on Diffusion-Time

Embodiments of the present invention use a measured diffusion time toadaptively control lithium surface concentration to keep thisconcentration below saturation. An important property governing thedynamics of the surface concentration is diffusion time τ. Many cellcharacteristics, such as graphite granularity, temperature, and averagelithium concentration can affect τ. To follow these changes, thedisclosed charging technique makes periodic measurements of τ. Incontrast, conventional charging profiles take a one-size-fits-allapproach to fixing the charge rate, do not adapt, and must assume theworst-case variability.

Note that measurements of τ and the cell current are effective aggregatevalues over all regions of an individual cell. That is the best that canbe done given that there are only two electrodes per cell (see FIG. 1).Localized variations that result in current density hot spots or locallyslow lithium transport cannot be addressed directly, since we do nothave local information to draw upon, only the behavior of the entirecell. Properties that are spatially uniform within a cell are readilyhandled by this technique. Properties that vary from one region toanother within a cell are measured in the aggregate. Hence, what ismeasured at the electrodes is neither the regional best τ nor the worstbut the average. Similarly, the measured current represents the averagecurrent density across the cell. The technique includes a parameter toaccount for the worst-case regional or spatial variation, which isderived from sampling a representative population of cells. This newcharging technique is a major improvement over past schemes in that onlyworst-case spatial variations within the cell must be characterizedrather than attempting to characterize the worst-case cell across theentire population of cells.

A diffusion equation governs the transport of lithium into the graphite.(It also arises in the modeling of heat conduction, particle flow, andother phenomena.) More specifically, a schematic representation oflithium transport into the graphite electrode is shown in FIG. 7, whichillustrates a lithium concentration profile u(X,t) through the graphiteelectrode. Note that the SEI is at X=0 and the copper current collectoris at some effective distance X=L. The lithium ions diffuse through theelectrolyte to the SEI layer, and they are reduced to metallic lithiumupon passing through this layer to the graphite, where they diffuse intothe graphite electrode and intercalate.

For a uniform slab with uniform boundary conditions, the diffusionequation for the lithium concentration u reduces to one spatialdimension. Expressed in terms of the dimensionless distance x=X/L, itbecomes

$\begin{matrix}{{\frac{\partial u}{\partial t} - {\frac{1}{\tau}\frac{\partial^{2}u}{\partial x^{2}}}} = 0} & (15)\end{matrix}$

The characteristic diffusion time is τ=L²/D, where D is the diffusivityof lithium in graphite. No lithium leaves the graphite near the coppercurrent collector so the proper boundary condition at x=1 is for thelithium flux density to go to zero. On the other hand, the flux densityat the graphite-SEI boundary at x=0 is proportional to the chargingcurrent. Each electron reduces one Li⁺ ion that then diffuses into thegraphite and intercalates, so with an appropriate choice ofconcentration units, the boundary conditions become

$\begin{matrix}{{- \left\lbrack {{\partial u}\text{/}{\partial x}} \right\rbrack_{x = 0}} = {{\frac{I\;\tau}{Q_{\max}}\left\lbrack {{\partial u}\text{/}{\partial x}} \right\rbrack}_{x = 1} = 0}} & (16)\end{matrix}$

Here, I is the charging current, and Q_(max) is the capacity of the cellin units of charge. This normalization gives u=1 throughout the graphitefor a fully charged cell. Since Q_(max) depends on the open-circuitvoltages corresponding to empty and full as well as on the cell design,the choice is arbitrary but convenient. Any initial condition may behandled, but without loss of generality we can start with[u] _(t=0)=0  (17)The solution to this system for a constant current starting at t=0 is

$\begin{matrix}{{u\left( {x,t} \right)} = {\frac{I\;\tau}{Q_{\max}}\left\lfloor {\frac{t}{\tau} - \frac{1}{6} + {\frac{1}{2}\left( {1 - x} \right)^{2}} - {\frac{2}{\pi^{2}}{\sum\limits_{k = 1}^{\infty}\;{\frac{1}{k^{2}}{\mathbb{e}}^{{- k^{2}}\pi^{2}t\text{/}\tau}\cos\;\left( {k\;\pi\; x} \right)}}}} \right\rfloor}} & (18)\end{matrix}$

FIG. 8 shows how the concentration evolves from early to later times.Eventually, the transient terms decay and leave a uniformly risingparabolic concentration profile. One can also handle the nonzero currentat the graphite-SEI interface using zero current boundary conditions butadding a source term that injects lithium just inside the region. Inthis case, the diffusion equation becomes

$\begin{matrix}{{\frac{\partial u}{\partial t} - {\frac{1}{\tau}\frac{\partial^{2}u}{\partial x^{2}}}} = {\frac{I(t)}{Q_{\max}}{{\delta(x)}.}}} & (19)\end{matrix}$

The δ(x) in the source term is a Dirac delta function centered at x=0.This term injects lithium at a normalized rate of I(t)/Q_(max)concentrated here. The solution has the form

$\begin{matrix}{{{u\left( {x,t} \right)} = {{B_{0}(t)} + {2{\sum\limits_{k = 1}^{\infty}\;{{B_{k}(t)}\cos\;\left( {k\;\pi\; x} \right)}}}}},} & (20)\end{matrix}$where the equations of motion for the coefficients are

$\begin{matrix}{{{+ {\frac{k^{2}\pi^{2}}{\tau}B_{k}}} = \frac{I(t)}{Q_{\max}}},{k = 0},1,2,{K.}} & (21)\end{matrix}$

The equations of motion are valid even if τ varies with time, which maybe the case if the diffusion coefficient changes with temperature orwith overall state of charge. Integration of Equation 21 for a constantτ and a current step at t=0 gives a solution equivalent to Equation 18,but without the limiting parabolic profile made explicit

$\begin{matrix}{{u\left( {x,t} \right)} = {{\frac{I}{Q_{\max}}t} + {\frac{I\;\tau}{Q_{\max}}\frac{2}{\pi^{2}}{\sum\limits_{k = 1}^{\infty}\;{{\frac{1}{k^{2}}\left\lbrack {1 - {\mathbb{e}}^{{- k^{2}}\pi^{2}t\text{/}\tau}} \right\rbrack}\;\cos\mspace{11mu}{\left( {k\;\pi\; x} \right).}}}}}} & (22)\end{matrix}$Short Time Behavior from Diffusion in an Infinite Half Space

The response of the concentration near one boundary over times Δt<<τ isindependent of the effects of the opposite boundary, sinceconcentrations near there do not have time to diffuse into the localregion. Essentially, only the lithium concentration within a lengthl=√{square root over (DΔt)} will have a significant effect. Therefore,the short time response for the concentration near one boundary may beobtained by considering diffusion in a layer where the graphiteelectrode has no opposite boundary.

Equation 19 still governs diffusion, but now there is only one explicitboundary condition, namely for zero current on the single boundary. Thesolution in this case has the form

$\begin{matrix}{{u\left( {x,t} \right)} = {2{\int_{0}^{\infty}{{B\left( {k,t} \right)}\cos\;({kx})\ \frac{\mathbb{d}k}{2\pi}}}}} & (23)\end{matrix}$

This is the continuum analog of Equation 20. Substitution into Equation19 and representing the source term in the same basis gives theequations of motion for the coefficients

$\begin{matrix}{{\frac{\partial{B\left( {k,t} \right)}}{\partial t} + {\frac{k^{2}}{\tau}{B\left( {k,t} \right)}}} = {\frac{I(t)}{Q_{\max}}.}} & (24)\end{matrix}$

For zero concentration initial condition and a constant current startingat t=0, the solution is

$\begin{matrix}{{B\left( {k,t} \right)} = {\left. {\frac{I\;\tau}{Q_{\max}}\frac{1 - {\mathbb{e}}^{{- k^{2}}t\text{/}\tau}}{k^{2}}}\Rightarrow{u\left( {x,t} \right)} \right. = {\frac{I\;\tau}{Q_{\max}}\left\lbrack {{\left( {{{erf}\left( {\frac{1}{2}\sqrt{\frac{\tau}{t}}x} \right)} - 1} \right)x} + {2\sqrt{\frac{t}{\pi\tau}}{\mathbb{e}}^{{- \frac{1}{2}}\frac{x^{2}}{2t\text{/}\tau}}}} \right\rbrack}}} & (25)\end{matrix}$

Here,

${{erf}\;(x)} = {\frac{2}{\sqrt{\pi}}{\int_{0}^{x}{{\mathbb{e}}^{- y^{2}}\ {\mathbb{d}y}}}}$is the so-called error function, which tends to 1 for large x. Theconcentration near the interface after the step is

$\begin{matrix}{{u\left( {x,t} \right)} = {{2\frac{I}{Q_{\max}}\sqrt{\frac{\tau}{\pi}}\sqrt{t}} - {\frac{I\;\tau}{Q_{\max}}x} + {O\;{\left( x^{2} \right).}}}} & (26)\end{matrix}$Here, O(x²) means additional terms of order x² and higher. As expected,the concentration gradient corresponds exactly to the source current. Ifthe current is interrupted, the concentration will relax, and thesolution of Equation 24 with this first order profile as the initialcondition gives the concentration near the interface after theinterruption as

$\begin{matrix}{{u\left( {x,t} \right)} = {u_{0} - {2\;\frac{I}{Q_{\max}}\sqrt{\frac{\tau}{\pi}}\sqrt{t}} + {O\left( x^{2} \right)}}} & (27)\end{matrix}$Here, u₀ is the surface concentration just before the interruption inthe current.Measuring Cell Diffusion Times

Equations 26 and 27 are the basis of a class of methods for measuringthe diffusion time τ from current step or pulse relaxation that areknown in the field collectively as “galvanostatic intermittent titrationtechniques” (GITT). This section develops and explains one practicalmethod.

The lithium concentration in the graphite u(x,t) can be estimated usingEquation 20 and Equation 21 by knowing the measured current I, the cellcapacity Q_(max) and τ. The diffusion time τ can be estimated, accordingto Equation 27, by calculating the relaxation of the lithium surfaceconcentration a short time after a current interruption. For Equation 27to be useful, however, what is needed is the relationship between therelaxation in surface concentration and the measured cell voltage.

Since charge must flow in an external circuit to do the work to transfera lithium atom from the positive electrode to the negative electrode,this work is observable as an electromotive force (EMF). Under relaxed,zero current conditions, the voltage measured across the cell equals thecell's EMF.

The work done for a reversible process under conditions of constanttemperature and pressure is given by the change in the Gibb's freeenergy:

$\begin{matrix}{{dG} = {- {\sum\limits_{i}\;{\mu_{i}{{dN}_{i}.}}}}} & (28)\end{matrix}$

Here, the work done is written in terms of the chemical potentials μ_(i)and numbers N_(i) of species i in the system. (Conceptually, theabsolute chemical potential is the work required to remove one unit of aspecies from the system and place it an infinite distance away.) For thelithium ion cell, the work done to transfer one lithium atom isΔG=−(μ_(Li,C) ₆ −μ_(Li,CoO) ₂ )  (29)where μ_(Li,C) ₆ and μ_(Li,CoO) ₂ are the chemical potentials for thelithium in the graphite and in the cobalt dioxide, respectively. Oneelectron of charge moves through the external circuit in this process,so the EMF is given by

$\begin{matrix}{E = {\frac{\Delta\; G}{q_{e}} = \frac{\mu_{{Li},{CoO}_{2}} - \mu_{{Li},C_{6}}}{q_{e}}}} & (30)\end{matrix}$where q_(e) is the elementary electronic charge. This can be expressedin terms of the electrochemical potentials ε_(Li,C) ₆ and εLi,CoO ₂ forthe respective half-cell reactionsLi_(x)C₆→Li_(x-y)C₆ +yLi⁺ +ye ⁻  (31)LiCoO₂→Li_(1-y)CoO₂ +yLi⁺ +ye ⁻  (32)asE=ε _(Li,CoO) ₂ −ε_(Li,C) ₆ .  (33)

The electrochemical potentials depend on the lithium concentrations inthe respective electrodes, lithium ion concentrations nearby in theelectrolyte, and so on. Practical cell formulations may incorporateadditional substances into the electrode materials that modify thechemistry somewhat, but the principles are the same. Taking diffusion oflithium into the graphite to be the rate-determining process, all otherconcentrations are treated as equilibrated. For example, thedistribution of ions in the electrolyte and of lithium in the cobaltdioxide is assumed to be uniform. Consequently, the positive electrodeelectrochemical potential ε_(Li,CoO) ₂ depends on the averageconcentration of lithium in the cobalt dioxide, which is simply relatedto the state of charge q, which does not change during the currentinterruption. The state of charge q is defined as the fraction ofremaining cell capacity ranging from empty (q=0%) to full (q=100%).

On the other hand, for the lithium in the graphite the electrochemicalpotential ε_(Li,C) ₆ depends on the slowly changing lithium surfaceconcentration. If the cell is given sufficient time to rest,approximately half of the diffusion time or τ/2, the distribution oflithium in the graphite becomes uniform and the lithium surfaceconcentration, with suitable normalization, equals the state of chargeq.

FIG. 9 shows the relaxed cell voltage (blue) as a function of the stateof charge. This voltage is calculated from the electrochemicalpotentials for the positive ε_(Li,CoO) ₂ (red) and negative ε_(Li,C) ₆(green) electrodes with respect to a lithium metal reference electrode(often referred to as the “Li/Li⁺” electrode in the field). The cellopen-circuit voltage after relaxation V_(OC,relax) is simply given bythe difference between the positive and negative electrochemicalpotentials:V _(OC,relax)(q)=ε_(Li,CoO) ₂ (q)−ε_(Li,C) ₆ (q).  (34)

The electrochemical potentials are fundamental properties of thechemistry of the cell; hence, the curves in FIG. 9 apply to all cellsconstructed with the same materials. The electrochemical potentials,ε_(Li,CoO) ₂ and ε_(Li,C) ₆ also depend upon the absolute temperature Taccording to the Nernst equation:

$\begin{matrix}{{ɛ = {ɛ^{o} - {\frac{k_{B}T}{{nq}_{e}}\ln\mspace{11mu} Q}}},} & (35)\end{matrix}$where ε° is the standard electrode potential for all reactants andproducts in their reference states, k_(B) is the Boltzmann constant, nis the number of electrons transferred in the half-reaction (n=1 forthis reaction), and Q is the reaction quotient. (Note that Q is theproduct of the relative activities of the products and reactants, eachraised to a power appropriate to the stoichiometry for the reaction; itis 1 for all materials in their reference states.)

Neglecting any small temperature dependence in Q, the Nernst equationcan be recast as a linear temperature correction to the voltage thatdepends only on the relevant concentration, such as the state of chargeq(t) for the positive electrode and the lithium surface concentrationu(0,t) for the negative electrode. The temperature correction factor ηis determined from measurements of the electrochemical potentials takenat 15° C., 25° C., 35° C., and 45° C. FIG. 10 shows the temperaturecorrection factors η for the positive and negative electrode potentialstaking T₀=25° C. as the reference point. At any other temperature, thecorrected potential isε(q,T)=ε(q,T ₀)+η(q)·(T−T ₀).  (36)

The open-circuit voltage, V_(OC), when not fully relaxed, can now berelated to the lithium concentrations and temperature byV _(OC)(t,T)=ε_(Li,CoO) ₂ (q,T)−ε_(Li,C) ₆ (v(t),T),  (37)designating v(t) as the lithium concentration at the surface of thegraphite as determined by the cell voltage. Measurement of therelaxation of the open circuit voltage V_(OC)(t,T) after a currentinterruption can now be used to estimate the lithium surfaceconcentration v(t) by using Equations 36 and 37 with the parameterizeddata in FIG. 9, which characterizes ε_(Li,C) ₆ (q,T₀) and ε_(Li,CoO) ₂(q,T₀), and the parameterized data in FIG. 10, which characterizesη_(Li,C) ₆ and η_(Li,CoO) ₂ . The estimate of the lithium surfaceconcentration v(t) from the relaxation of the open circuit voltageV_(OC)(t,T) can be used with the derivative of Equation 27 to calculateτ:

$\begin{matrix}{{\frac{\mathbb{d}{v(t)}}{\mathbb{d}\sqrt{t}} = {{- 2}\;\frac{I}{Q_{\max}}\sqrt{\frac{\tau}{\pi}}}},} & (38)\end{matrix}$where I is the constant current that was interrupted.

Several different dynamic processes occur in a cell with nonzerocurrent, one of which is the slow diffusion examined here. The othersact over much shorter time scales (less than one second), and their neteffect to a constant current is to contribute a resistive-like potentialdrop IR. When the constant current is interrupted in order to useEquation 38 to measure τ, this potential drop disappears, leaving theslow relaxation of the open circuit voltage V_(OC)(t,T), given byEquation 37.

FIG. 11 shows the cell voltage versus time for a 7.5 minute, 0.4 Cdischarge (about 1.0 A for this cell) followed by one hour ofrelaxation. In far less than a second after the discharge is stopped, aresistive drop of about 70 mV disappears and the long relaxation fromthe changing lithium surface concentration is seen.

Equation 38 indicates that the lithium surface concentration v(t) shouldbe linear with respect to the square root of time for short time periodswith a slope given by the square root of the diffusion time τ. FIG. 12shows a plot of the estimated lithium surface concentration v(t)obtained from the relaxation of the open circuit voltage after aconstant current interruption versus the square root of time as measuredfrom the end of the discharge pulse. The behavior for short times islinear as predicted, and the diffusion time obtained from the slope is≈2070 s. Qualitative deviations from the square root of time behaviorare seen in a plot of the full analytic solution for the surfaceconcentration for t≧0.02τ, so based on the data shown in FIG. 12, thereal diffusion time may be greater than 3200 s.

We measured τ versus state of charge for twenty cells at 15° C., 25° C.,35° C., and 45° C. The results are presented in FIG. 13. The diffusiontime varies as expected with temperature, namely diffusion is slower forthe cold cells than for the warm ones. There is also a consistent peakin the diffusion time in an intermediate range of the state of charge,which suggests that this is the condition where the lithium surfaceconcentration may saturate when charging.

Ideal Charging

The ideal charging technique brings the lithium surface concentration to100% and holds it there until the lithium concentration throughout allof the graphite is also 100%, indicating a fully charged cell. To findthe charging current, the diffusion equation, Equation 15, is solvedwith the boundary conditions being that the surface (x=0) concentrationis 100% and that no lithium leaves the graphite near the copper currentcollector (x=1)[u] _(u=0)=1[

u/

x] _(x=1)=0.  (39)Applying the initial condition [u]_(t=0)=0, the solution to thediffusion equation is given by,

$\begin{matrix}{{{u\;\left( {x,t} \right)} = {1 + {2\;{\sum\limits_{k = 1}^{\infty}\;{{B_{k}(t)}\;{\sin\mspace{11mu}\left\lbrack {\left( {k - \frac{1}{2}} \right)\pi\; x} \right\rbrack}}}}}},{where}} & (40) \\{{B_{k}(t)} = {\frac{- 1}{\pi\left( {k - \frac{1}{2}} \right)}{{\mathbb{e}}^{{- {({k - \frac{1}{2}})}^{2}}\pi^{2}t\text{/}\tau}.}}} & (41)\end{matrix}$The ideal charging current is obtained by applying to the solution therelationship between the slope of the concentration at the surface andthe current

$\begin{matrix}{{- \left\lbrack {{\partial u}\text{/}{\partial x}} \right\rbrack_{x = 0}} = {\frac{I\;\tau}{Q_{\max}}.}} & (42)\end{matrix}$By taking the derivative of Equation 38 with respect to x at x=0 andsubstituting in Equation 42, the solution to the ideal charging currentis

$\begin{matrix}{{I_{ideal}(t)} = {2\;\frac{Q_{\max}}{\tau}{\sum\limits_{k = 1}^{\infty}\;{{\mathbb{e}}^{{- {({k - \frac{1}{2}})}^{2}}\pi^{2}t\text{/}\tau}.}}}} & (43)\end{matrix}$For short times Δt<<τ and short distances l=√{square root over(DΔt)}<<L, the presence of the opposite boundary can again be neglected,allowing for the solution to the diffusion equation in an infinitehalf-space. The solution has the form similar to Equation 40, butcontinuous as expected

$\begin{matrix}{{u\left( {x,t} \right)} = {1 + {2\;{\int_{0}^{\infty}{{B\left( {k,t} \right)}\mspace{11mu}\sin\mspace{11mu}({kx})\ {\frac{\mathbb{d}k}{2\pi}.}}}}}} & (44)\end{matrix}$Substituting into the diffusion equation, Equation 15, and applying theinitial condition [u]_(t=0)=0 yields

$\begin{matrix}{{{u\left( {x,t} \right)} = {1 - {{erf}\left\lfloor {\frac{1}{2}\frac{x}{\sqrt{t/\tau}}} \right\rfloor}}},} & (45)\end{matrix}$where again

${{erf}(x)} = {\frac{2}{\sqrt{\pi}}{\int_{0}^{x}{e^{- y^{2}}{{\mathbb{d}y}.}}}}$Using the relation given in Equation 42, the ideal short time chargingcurrent is then

$\begin{matrix}{{I_{deal}(t)} = {\frac{Q_{\max}}{\sqrt{\pi\tau}}{\frac{1}{\sqrt{t}}.}}} & (46)\end{matrix}$

Any practical application of a current of this form would use theaverage required current over some time interval, Δt

$\begin{matrix}{\left\langle I_{ideal} \right\rangle_{\Delta\; t} = {2\frac{Q_{\max}}{\sqrt{\pi\tau}}{\frac{1}{\sqrt{\Delta\; t}}.}}} & (47)\end{matrix}$Note that the long time behavior of the ideal charging current isgoverned by the k=1 term of Equation 43, where the current decaysexponentially with a time constant of 4τ/π² or about 0.4τ. The shorttime behavior is given by Equation 46, in which the current decays asthe square root of time.

FIG. 14 shows a simulation of ideal charging with a constant diffusiontime τ. The top two plots show the charging current versus time: the topplot is on a log-log scale, while the second plot is on a log-linearscale. The top plot illustrates the dependence of the charging currenton the square root of time for times less than about 0.2τ. The secondplot shows the exponential behavior of the charging current for timesgreater than about 0.4τ. Note that the transition from square root timebehavior to exponential behavior for the charging current occurs between0.2τ and 0.4τ. The third plot in FIG. 14 shows the state of chargeversus time in units of the diffusion time τ, indicating that underideal conditions, a cell can be fully charged in about 1.5τ. The finalplot shows the lithium concentration as a function of the distanceacross the graphite from the SEI for eight equally spaced times duringcharging. Notice how the boundary condition of 100% lithiumconcentration at the surface, Equation 39, is maintained throughout theideal charge.

In a real system τ is not constant and depends upon the state of charge,the temperature, and other cell characteristics. FIG. 15 shows the idealcharging profile with an always known, but changing, τ. Morespecifically, FIG. 15 shows the charging current, τ, and the state ofcharge as a function of time. The τ used in the simulation comes from aparameterization of the diffusion time data in FIG. 13 for a slower thantypical cell at 14° C. Notice that the ideal charge tends to varyinversely with the diffusion time, slowing considerably, in thisexample, at a state of charge between 50% and 70%, where τ reachesnearly 30000 seconds. Notice also that above a state of charge of 70% atabout 1.2 hours, the charging current increases as 2 decreases. Thissimulation indicates how a charging technique that takes advantage ofthe diffusion time to maintain the lithium surface concentration at orbelow 100% can charge a cell very quickly (2.5 hours) even in a worstcase scenario (slow cell at 14° C.).

Diffusion-Limited Adaptive Charging

An ideal adaptive charging technique adjusts the charging current tomaintain the lithium surface concentration at 100%, but assumes that τis known at all times. In contrast, Diffusion-Limited Adaptive Chargingis a practical charging technique that calculates the lithium surfaceconcentration based upon periodic measurements of τ and calculates acharging current that keeps the lithium surface concentration at orbelow 100%, preventing graphite saturation at the separator interface.The optimal charging current I_(opt) that maintains the lithium surfaceconcentration at 100% is given by Equation 27. We apply the results ofthe infinite half-space problem for an iteration time, t_(n+1)−t_(n),which is significantly faster than τ:

$\begin{matrix}{{I_{opt}\left( t_{n + 1} \right)} = {2\frac{Q_{\max}}{\sqrt{\pi\tau}}{\frac{\left( {1 - {u_{I = 0}\left( {0,t_{n + 1}} \right)}} \right)}{\sqrt{t_{n + 1} - t_{n}}}.}}} & (48)\end{matrix}$u_(I=0)(0,t_(n+1)) is the lithium surface concentration calculated fromthe diffusion model with the current shut off (I=0) for the nextiteration time. Equation 48 gives the current required to compensate forthe relaxation that would occur with zero current.

The zero current lithium surface concentration u_(I=0)(0,t_(n+1)) isestimated from Equations 20 and 21. Equation 20 gives the surfaceconcentration from the diffusion model from a set of evolvingcoefficients B_(k)(t):

$\begin{matrix}{{u\left( {0,t_{n}} \right)} = {{B_{0}\left( t_{n} \right)} + {2{\sum\limits_{k = 1}^{\infty}{{B_{k}\left( t_{n} \right)}.}}}}} & (49)\end{matrix}$Looking one time iteration ahead with zero current, it is the samerelationship with coefficients B_(k,I=0)(t) that have been evolved onestep with zero current:

$\begin{matrix}{{u_{I = 0}\left( {0,t_{n + 1}} \right)} = {{B_{0,{I = 0}}\left( t_{n + 1} \right)} + {2{\sum\limits_{k = 1}^{\infty}{{B_{k,{I = 0}}\left( t_{n + 1} \right)}.}}}}} & (50)\end{matrix}$The evolution of the coefficients B_(k)(t) in Equation 21 can becalculated in an iterative and stable fashion by using the forwarddifference (or implicit difference) to approximate the derivative ofB_(k)(t):

$\begin{matrix}{{B{{\overset{\prime}{Y}}_{k}\left( t_{n} \right)}} = {\frac{{B_{k}\left( t_{n} \right)} - {B_{k}\left( t_{n - 1} \right)}}{t_{n} - t_{n - 1}}.}} & (51)\end{matrix}$Using the forward difference calculation, the B_(k) coefficients areaccurate for time steps given by:

$\begin{matrix}{\left( {t_{n} - t_{n - 1}} \right) < {\frac{\tau}{k^{2}\pi^{2}}.}} & (52)\end{matrix}$

Note that for a time step of 1 second and a minimum diffusion time of3200 seconds, only the first 18 B_(k) terms are accurate. In contrast,for a τ of 50000 seconds, the first 71 B_(k) terms are accurate. Thebenefit of using the forward difference is that, although the estimatesmay be inaccurate, the higher order terms go toward zero alwaysproducing a stable net result. Also, the higher order terms (even ifinaccurate) are often insignificant and can be ignored without anysignificant error on the overall estimation.

Substituting Equation 51 into Equation 21, one can solve forB_(k)(t_(n)) based on the measured current I normalized to totalcapacity Q_(max), the diffusion time τ, and the previous value ofB_(k)(t_(n−1)):

$\begin{matrix}{{B_{k}\left( t_{n} \right)} = {\frac{{B_{k}\left( t_{n - 1} \right)} + {\left( {t_{n} - t_{n - 1}} \right)\frac{I\left( t_{n} \right)}{Q_{\max}}}}{1 + {\left( {t_{n} - t_{n - 1}} \right)\frac{k^{2}\pi^{2}}{\tau}}}.}} & (53)\end{matrix}$

To calculate the predicted lithium surface concentration, assuming zerocurrent for the next iteration, simply calculate the next iterationB_(k,I=0)(t_(n+1)) assuming a future current I(t_(n+1)) of zero:

$\begin{matrix}{{B_{k,{I = 0}}\left( t_{n + 1} \right)} = {\frac{B_{k}\left( t_{n} \right)}{1 + {\left( {t_{n + 1} - t_{n}} \right)\frac{k^{2}\pi^{2}}{\tau}}}.}} & (54)\end{matrix}$Equation 48 may now be used to calculate the optimal charging currentI_(opt) assuming that the diffusion time τ is known.

Since τ varies with temperature, state of charge, and other cellvariations (see FIG. 13), always knowing the diffusion time accuratelyis problematic. In Diffusion-Limited Adaptive Charging, τ is re-measuredfrequently, such as every few minutes, using the voltage relaxationtechnique described earlier, since the cell temperature or state ofcharge cannot change significantly in such short periods. By measuringthe diffusion time frequently, corrections to the diffusion time basedon changes to the temperature or state of charge are unnecessary.

To measure the diffusion time of a cell, the charging current is set tozero periodically to measure the relaxation of the open circuit voltage.The open-circuit voltage V_(OC), state of charge q, and measuredtemperature T can be used to estimate the lithium surface concentrationv(t) using Equation 37 for two times following the current interruption(t1 and t2). These two estimates of the lithium surface concentrationv(t1) and v(t2) are combined with Equation 38 to determine the diffusiontime:

$\begin{matrix}{\tau = {\pi\left\lbrack {\frac{{v\left( {t\; 2} \right)} - {v\left( {t\; 1} \right)}}{\sqrt{t\; 2} - \sqrt{t\; 1}}\frac{Q_{\max}}{2I}} \right\rbrack}^{2}} & (55)\end{matrix}$

Equation 38 is valid for a relaxation period t_(rest) that follows aconstant current period t_(cc) at least as long as the relaxation time.Also, Equation 38 is valid only for a relaxation time t_(rest) muchshorter than τ. Specifically, for a fixed relaxation time t_(rest), themeasured τ is valid only if it is larger than the relaxation timedivided by about 0.02. To be conservative during charging, the diffusiontime τ_(m) is equal to the minimum of the measured τ and the minimumaccurate diffusion time for the fixed relaxation time.

$\begin{matrix}{\tau_{m} = {\min\left( {\tau,\frac{t_{r}}{0.02}} \right)}} & (56)\end{matrix}$

For example, with a 64 second relaxation time, the current beforerelaxation should be constant for at least 64 seconds, and diffusiontimes less than about 3200 seconds cannot be accurately measured. Thecell voltages V_(OC)(t1) and V_(OC)(t2) needed to measure τ could bemeasured at t1=4 seconds and t2=t_(rest)=64 seconds. By reducing therelaxation time, one increases the ability to measure smaller diffusiontimes, but the smaller relaxation time places higher requirements on thecell voltage precision, as the voltage relaxation is smaller for ashorter relaxation period.

Since τ depends upon temperature T and state of charge q, which changeduring charging, the diffusion time should be measured often. The periodof time between diffusion time measurements t_(p) is sum of the time theoptimal charging current is applied t_(opt), the relaxation timet_(rest), and the constant current time t_(cc):t _(p) =t _(opt) +t _(rest) +t _(cc).  (57)

To account for the possibility of a changing τ between the periodicmeasurements or inaccuracies in using the cell voltage to measure τ, thediffusion time can be arbitrarily increased by a conservative factor α.The larger this conservative factor α, the more slowly the cell willcharge, because it is charged as if its diffusion time was longer thanactually measured.

The conservative factor α must also account for the worst-casenon-uniformity in cells arising, for instance, from current density hotspots or locally slow diffusion times. For instance, if the currentdensity in one local spot of the cell is 50% higher than the averagecurrent density, the conservative factor α would need to be at least2.25, since τ varies as the current squared, as seen in Equation 55.

The conservative factor α is required to be greater than 1 with atypical value of 2, and can be optimized for any specific implementationor worst-case current or diffusion time non-uniformity.τ_(c)=α·τ_(min).  (58)

In calculating the projected, zero current surface concentrationu_(I=0)(0,t_(n+1)) and the optimal charging current I_(opt), theconservative τ_(c) should be used instead of τ in Equations 48, 53 and54.

If charging always began with a rested cell at zero state of charge anda known diffusion time τ_(c), then the B_(k) terms could simply beinitialized to zero at the start of charge, and evolved from the initialstate using a measured τ_(c) and current I. Often, however, thedistribution of lithium in the graphite u(x,t) and the diffusion timeτ_(c) are unknown at the start of charging. Even if τ_(c) was known,u(x,t) cannot be accurately calculated over long periods of time due tothe random walk nature of integrating the current I(t) that has a whitenoise component. This is the same problem that occurs with coulombcounting to determine the state of charge q(t), where the uncertaintygrows as the square root of time.

In order to initialize the B_(k) parameters at the start of charge, thelithium distribution needs to be in a known state and τ_(c) must bemeasured. Diffusion-Limited Adaptive Charging addresses this problem bycharging the cell at a low constant current h for a sufficient period oftime t_(init), such as charging with 0.1 C for 5 minutes, so that thedistribution of lithium in the graphite can be described by theparabolic distribution given in Equation 18. After the constantinitialization current, the current is set to zero and the cell isallowed to relax for t_(rest) seconds, and the first measurement ofdiffusion time τ_(c0) can be measured using Equation 58.

Using Equation 20 and Equation 22, one can solve for the B_(k) terms atthe surface (x=0) for a constant charging current I₀ for a time t_(o)with a diffusion time of τ_(c0).

$\begin{matrix}{{B_{0}\left( t_{0} \right)} = {\frac{I_{0}t_{init}}{Q_{\max}} = q}} & (59) \\{{B_{k}\left( t_{init} \right)} = {{{\frac{I_{0}\tau_{c\; 0}}{Q_{\max}k^{2}\pi^{2}}\left\lbrack {1 - e^{{- k^{2}}\pi^{2}{t_{init}/\tau_{c\; 0}}}} \right\rbrack}k} \geq 1}} & (60)\end{matrix}$Note that B₀(t), i.e. when k=0, is the integral of the current Inormalized to Q_(max), which is simply equal to the state of charge q.

Equation 59 and Equation 60 describe the B_(k) terms at the end of theconstant current I₀ charge, but not at the end of the initial relaxationperiod. Since the current is zero during the τ_(c0) measurement period,B₀(t_(init)) is equal to B₀(t_(init)+t_(rest)), which is the state ofcharge q. To determine the other B_(k) terms at the end of the initialrelaxation period t_(init)+t_(rest), Equation 60 is iterated usingEquation 54, resulting in:

$\begin{matrix}{{{B_{k}\left( {t_{init} + t_{rest}} \right)} = {\frac{I_{0}\tau_{c\; 0}}{Q_{\max}k^{2}\pi^{2}}\frac{\left\lbrack {1 - e^{{- k^{2}}\pi^{2}{t_{init}/\tau_{c\; 0}}}} \right\rbrack}{\left\lbrack {1 + {\left( {t_{n + 1} - t_{n}} \right)\frac{k^{2}\pi^{2}}{\tau_{c\; 0}}}} \right\rbrack^{\frac{t_{rest}}{({t_{n + 1} - t_{n}})}}}}},{k \geq 1}} & (61)\end{matrix}$

After initializing the B_(k) terms at the end of the initial calibrationcharge using Equation 59 and Equation 61, the B_(k) terms can thereafterbe iteratively updated using Equation 53 with the measured current I,and the most recent diffusion time τ_(c).

During each time iteration, the optimal charging currentI_(opt)(t_(n+1)) is calculated, but not always used. For instance, theremay be a maximum charging limit imposed by the charger, or a thermallimit, or a quantization of the charger's current set-point. Also, inorder to periodically measure the diffusion time τ_(c), the current isrequired to be constant for a period t_(cc), and zero for a relaxationperiod t_(rest), as discussed for Equation 55. Even when the optimalcharging current from Equation 48 is ignored for the above reasons, theB_(k) terms continue to be accurate, since they are updated using themeasured charging current I(t_(n)), regardless of the optimal chargingcurrent I_(opt)(t_(n)).

A simulation of charging a battery with Diffusion-Limited AdaptiveCharging is illustrated in FIG. 16. The charging current I(t), lithiumsurface concentration u(0,t), and state of charge q(t) were simulated byusing a parameterization of the diffusion time τ from the data shown inFIG. 13 as a function of temperature T and state of charge q. To accountfor variation in diffusion times between cells, the parameterizeddiffusion time τ is adjusted by ±20% to characterize slow (+20%),typical (+0%), and fast (−20%) cells.

The charging begins with an initialization current I₀/Q_(max) of 0.1 Cfor t₀=256 seconds, followed by the first diffusion time measurementτ_(m) after a relaxation of t_(r)=64 seconds. A conservative factor α of1.8 is used according to Equation 58 to obtain a conservative diffusiontime τ_(c). After the initialization current, the B_(k) terms areinitialized using Equation 59 and Equation 61 for k from 0 to 10, whilethe higher order k terms are ignored.

Thereafter, the B_(k) terms are updated iteratively everyt_(n)−t_(n−1)=1 second using Equation 53, and an optimal chargingcurrent is calculated using Equation 48. The normalized optimal currentI_(opt)/Q_(max) is limited to 0.7 C to simulate a maximum charging limitI_(max), and quantized to 128 mA to simulate the quantization of acharger's current set-point. For the first 192 seconds after arelaxation period, the charger is set to the optimal currentI_(opt)(t_(n+1)). For the next 64 seconds, the current is held constantbefore the relaxation period t_(r), which also lasts for 64 seconds. Atthe end of each relaxation period, the diffusion time τ_(c) is measured.This cycle repeats until the cell is charged.

FIG. 16 shows simulation results from charging a cell at two differenttemperatures: 14° C. on the left and 25° C. on the right. The top plotsin the figure show the charging current I/Q_(max) as a function of time,where the periodic resting periods to measure τ_(c) can be seen followedby current surges. The current surges following a rest period arisenaturally with the technique to make up for the decrease in lithiumsurface concentration during the rest period, when the lithium surfaceconcentration has time to relax. The middle plots show the truediffusion time τ (blue) as well as the diffusion time measured at theend of each relaxation period (green). The plot on the bottom shows thestate of charge q (blue), the lithium surface concentration estimator(red) using the sampled diffusion rate τ increased by the conservativefactor α, and the true lithium surface concentration (green) using thetrue diffusion rate τ. The simulation shows that the estimated time tocharge a cell at 14° C. is about 4 hours, while a cell at 25° C. chargesin less than 3 hours.

Battery Design

FIG. 17 illustrates a rechargeable battery 1700 that supports adaptivecharging in accordance with an embodiment of the present invention.Battery 1700 includes a battery cell 1702, which is illustrated in moredetail in FIG. 1. It also includes a current meter (current sensor)1704, which measures a charging current applied to cell 1702, and avoltmeter (voltage sensor) 1706, which measures a voltage across cell1702. Battery 1700 also includes a thermal sensor 1730, which measuresthe temperature of battery cell 1702. (Note that numerous possibledesigns for current meters, voltmeters and thermal sensors arewell-known in the art.)

Rechargeable battery 1700 also includes a current source 1723, whichprovides a controllable constant charging current (with a varyingvoltage), or alternatively, a voltage source 1724, which provides acontrollable constant charging voltage (with a varying current).

The charging process is controlled by a controller 1720, which receives:a voltage signal 1708 from voltmeter 1706, a current signal 1710 fromcurrent meter 1704 a temperature signal 1732 from thermal sensor 1730and a state of charge (SOC) value 1732 from SOC estimator 1730. Theseinputs are used to generate a control signal 1722 for current source1723, or alternatively, a control signal 1726 for voltage source 1724.

During operation, SOC 1732 estimator receives a voltage 1708 fromvoltmeters 1706, a current from current meter 1704 and a temperaturefrom thermal sensor 1730 and outputs a state of charge value 1732. (Theoperation of SOC estimator 1730 is described in more detail below.)

Note that controller 1720 can be implemented using either a combinationof hardware and software or purely hardware. In one embodiment,controller 1720 is implemented using a microcontroller, which includes amicroprocessor that executes instructions which control the chargingprocess.

The operation of controller 1720 during the charging process isdescribed in more detail below.

Charging Process

FIG. 18 presents a flow chart illustrating the charging process inaccordance with an embodiment of the present invention. At a high level,the system first determines a lithium surface concentration at aninterface between the transport-limiting electrode and the electrolyteseparator (step 1802). Next, the system uses the determined lithiumsurface concentration to control a charging process for the battery sothat the charging process maintains the lithium surface concentrationwithin set limits (1804).

In a more specific embodiment, referring to FIG. 19, the system firstdetermines a potential of the transport-limiting electrode with respectto a known reference, wherein the potential is correlated with thelithium surface concentration (step 1902). Next, the system uses thedetermined potential of the transport-limiting electrode in a controlloop, which adjusts either a charging voltage or a charging current, tomaintain the potential of the transport-limiting electrode at a levelwhich keeps the lithium surface concentration within set limits (step1904).

Determining the Potential of the Transport-limiting Electrode

FIG. 20 presents a flow chart illustrating the process of determining apotential of a transport-limiting electrode with respect to a knownreference in accordance with an embodiment of the present invention. Inthis embodiment, the system measures the temperature of the batterythrough a thermal sensor (step 2002). The system also measures a currentthrough the battery (step 2004) and a total cell voltage of the battery(between the electrodes) (step 2006).

The system also determines a state of charge of the battery (step 2008).In one embodiment of the present invention, this involves reading thestate of charge from a “gas gauge integrated circuit,” such as partnumber bq27000 distributed by Texas Instruments of Dallas, Tex. Thesegas gauge circuits generally operate by determining a state of charge ofthe battery from a previous state of charge of the battery in additionto a measured current, a measured temperature and a measured total cellvoltage of the battery.

Next, the system determines the potential of the non-transport-limitingelectrode (generally the positive electrode) from the determined stateof charge and the temperature (step 2010). Finally, the systemdetermines the potential of the transport-limiting electrode (generallythe negative electrode) by starting with the measured total cell voltageand subtracting the determined potential of the non-transport-limitingelectrode, and also subtracting a voltage drop caused by the measuredcurrent multiplied by a resistance through the battery (step 2012).

Charging Process Based on Diffusion Time

FIG. 21 presents a flow chart illustrating a charging process based onmeasuring the diffusion time τ for a transport-limiting electrodegoverned by diffusion in accordance with an embodiment of the presentinvention. In this embodiment, the system first measures a diffusiontime τ for lithium in the negative electrode (step 2102). (This caninvolve using a process which is described in more detail below withreference to FIG. 22.) Next, the system estimates the lithium surfaceconcentration based on the diffusion time τ, a cell capacity Q_(max) forthe battery and a measured charging current I for the battery (step2104).

In one embodiment, the system measures τ periodically, and this measuredvalue for τ is used to model how the surface concentration evolvesbetween τ measurements based on the charging current I and the cellcapacity Q_(max). For example, τ can be measured every few minutes, andthis measured τ value can be used in a model for the surfaceconcentration, which is updated every second between τ measurements.

Next, the system calculates a charging current or a charging voltage forthe battery based on the estimated lithium surface concentration (step2106), which can involve performing the calculation in Equation 48.Next, the system applies the calculated charging current or chargingvoltage to the battery, for example through current source 1723 orvoltage source 1724 (step 2108).

Process of Measuring Diffusion Time

FIG. 22 presents a flow chart illustrating the process of measuring thediffusion time τ in accordance with an embodiment of the presentinvention. During this process, the system first charges the batterywith a fixed current for a fixed time period (step 2202). Next, thesystem enters a zero current state in which the charging current is setto zero (step 2204). During this zero current state, the system measuresan open circuit voltage for the battery at two times while the opencircuit voltage relaxes toward a steady state (step 2206). Finally, thesystem calculates the diffusion time τ based on the measured opencircuit voltages (step 2208), for example, using Equations 37 and 55.

The foregoing descriptions of embodiments have been presented forpurposes of illustration and description only. They are not intended tobe exhaustive or to limit the present description to the formsdisclosed. Accordingly, many modifications and variations will beapparent to practitioners skilled in the art. Additionally, the abovedisclosure is not intended to limit the present description. The scopeof the present description is defined by the appended claims.

What is claimed is:
 1. A method for adaptively charging a battery,wherein the battery is a lithium-ion battery that includes atransport-limiting electrode governed by diffusion, an electrolyteseparator and a non-transport-limiting electrode, comprising while thebattery is charging: charging the battery with a fixed current for afixed time period; entering a zero current state in which the chargingcurrent is set to zero; during the zero current state, measuring an opencircuit voltage for the battery at two times while the open circuitvoltage relaxes toward a steady state; and calculating a chargingcurrent or a charging voltage based on the measured open circuitvoltages; and applying the calculated charging current or the calculatedcharging voltage to the battery.
 2. The method of claim 1, whereincalculating the charging current or the charging voltage based on themeasured open circuit voltages includes: calculating a diffusion time Tbased on the measured open circuit voltages; calculating a lithiumsurface concentration based on the diffusion time T, a state of chargeof the battery q and a temperature of the battery T; and calculating thecharging current or the charging voltage for the battery based on thecalculated lithium surface concentration.
 3. The method of claim 1,wherein the transport-limiting electrode is a negative electrode; andwherein the non-transport-limiting electrode is a positive electrode. 4.The method of claim 3, wherein the negative electrode is comprised ofgraphite and/or TiS₂; wherein the electrolyte separator is a liquidelectrolyte comprised of LiPF₆, LiBF₄ and/or LiClO₄ and an organicsolvent; and wherein the positive electrode is comprised of LiCoO₂,LiMnO₂, LiFePO₄ and/or Li₂FePO₄F.
 5. A lithium-ion battery with anadaptive charging mechanism, comprising: a transport-limiting electrodegoverned by diffusion; an electrolyte separator; anon-transport-limiting electrode; a current sensor configured to measurea charging current for the battery; a voltage sensor configured tomeasure a voltage across terminals of the battery; a charging sourceconfigured to apply a charging current or a charging voltage to thebattery; a controller configured to receive inputs from the currentsensor and the voltage sensor, and configured to send a control signalto the charging source; wherein, while the battery is charging, thecontroller is configured to: charge the battery with a fixed current fora fixed time period, enter a zero current state in which the chargingcurrent is set to zero, during the zero current state, measure an opencircuit voltage for the battery at two times while the open circuitvoltage relaxes toward a steady state, and calculate a charging currentor a charging voltage based on the measured open circuit voltages, andapply the charging current or the charging voltage to the battery. 6.The lithium-ion battery of claim 5, wherein while calculating thecharging current or the charging voltage, the controller is configuredto: calculate a diffusion time τ based on the measured open circuitvoltages; calculate a lithium surface concentration based on thediffusion time τ, a state of charge of the battery q and a temperatureof the battery T; and calculate the charging current or the chargingvoltage for the battery based on the calculated lithium surfaceconcentration.
 7. The lithium-ion battery of claim 5, wherein thetransport-limiting electrode is a negative electrode; and wherein thenon-transport-limiting electrode is a positive electrode.
 8. Thelithium-ion battery of claim 7, wherein the negative is comprised ofgraphite and/or TiS₂; wherein the electrolyte separator is a liquidelectrolyte comprised of LiPF₆, LiBF₄ and/or LiClO₄ and an organicsolvent; and wherein the positive electrode is comprised of LiCoO₂,LiMnO₂, LiFePO₄ and/or Li₂FePO₄F.
 9. A non-transitory computer-readablestorage medium storing instructions that when executed by a controllerfor a battery cause the controller to perform a method for adaptivelycharging a battery, wherein the battery is a lithium-ion battery whichincludes a transport-limiting electrode governed by diffusion, anelectrolyte separator and a non-transport-limiting electrode, the methodcomprising, while the battery is charging: charging the battery with afixed current for a fixed time period; entering a zero current state inwhich the charging current is set to zero; during the zero currentstate, measuring an open circuit voltage for the battery at two timeswhile the open circuit voltage relaxes toward a steady state; andcalculating a charging current or a charging voltage based on themeasured open circuit voltages; and applying the charging current or thecharging voltage to the battery.
 10. The non-transitorycomputer-readable storage medium of claim 9, wherein calculating thecharging current or the charging voltage based on the measured opencircuit voltages includes: calculating a diffusion time τ based on themeasured open circuit voltages; calculating a lithium surfaceconcentration based on the diffusion time τ, a state of charge of thebattery q and a temperature of the battery T; and calculating thecharging current or the charging voltage for the battery based on thecalculated lithium surface concentration.
 11. A method for adaptivelycharging a battery, wherein the battery is a lithium-ion battery whichincludes a transport-limiting electrode governed by diffusion, anelectrolyte separator and a non-transport-limiting electrode,comprising: determining a lithium surface concentration at an interfacebetween the transport-limiting electrode and the electrolyte separatorbased on a diffusion time for lithium in the transport-limitingelectrode; calculating a charging current or a charging voltage for thebattery based on the determined lithium surface concentration; andapplying the charging current or the charging voltage to the battery.12. The method of claim 11, wherein the charging current or chargingvoltage is calculated to maintain the lithium surface concentrationwithin set limits.
 13. The method of claim 11, wherein calculating thecharging current or charging voltage involves maximizing the chargingcurrent or charging voltage while maintaining the lithium surfaceconcentration within the set limits.
 14. The method of claim 11, whereindetermining the lithium surface concentration involves: measuring adiffusion time τ for lithium in the transport-limiting electrode; andestimating the lithium surface concentration between τ measurementsbased on the diffusion time τ, a cell capacity Q_(max) for the batteryand a measured charging current I for the battery.
 15. The method ofclaim 14, wherein measuring the diffusion time τ involves periodically:charging the battery with a fixed current for a fixed time period;entering a zero current state in which the charging current is set tozero; during the zero current state, measuring an open circuit voltagefor the battery at two times while the open circuit voltage relaxestoward a steady state; and calculating the diffusion time τ based on themeasured open circuit voltages.
 16. The method of claim 15, wherein thetransport-limiting electrode is a negative electrode; and wherein thenon-transport-limiting electrode is a positive electrode.
 17. The methodof claim 16, wherein the negative electrode is comprised of graphiteand/or TiS₂; wherein the electrolyte separator is a liquid electrolytecomprised of LiPF₆, LiBF₄ and/or LiClO₄ and an organic solvent; andwherein the positive electrode is comprised of LiCoO₂, LiMnO₂, LiFePO₄and/or Li₂FePO₄F.